I am presented with the following equation system
$x^2 = 8+(y-z)^2$
$y^2 = 12+(z-x)^2$
$z^2 = 24+(x-y)^2$
I don't even know where to start. I am assuming you need to isolate the variables to begin with but even then - I have been unable to reach any sort of progress. Please help. Thank you.
It's $$(x-y+z)(x-z+y)=8$$ $$(y-x+z)(y-z+x)=12$$ and $$(z-x+y)(z-y+x)=24,$$ which gives $$\prod_{cyc}(x+y-z)^2=48^2.$$ Now, if $\prod\limits_{cyc}(x+y-z)=48$ we obtain: $$y+z-x=6,$$ $$x+z-y=4$$ and $$x+y-z=2,$$ which gives $$(x,y,z)=(3,4,5).$$ The second case gives $$(x,y,z)=(-3,-4,-5).$$